You should be fine walking at that time. I finished my camino (camino frances) this year in late May, and I had a bed to sleep in every night. Some days I had to walk quickly or stay in a casa rural, but finding a place to sleep won't be an issue. Especially when you are going- I think not as many people start the camino in the beginning of April, because there is still some rough weather in the mountains at that point (a blizzard in mid april between st jean and roncesvalles this past year, if I remember correctly). In addition to reading what is ahead in your guide, ask around to see if there are places that have limited spaces- sometimes albergues are still closed that time of year or are closed for other reasons (renovations, etc) and can surprise you when you are expecting lots of beds. I don't think you'll have any problems though. Arriving at the beginning of may would probably make it less crowded for you, but it is better to take your time and not rush, if possible....
buen camino! I wish I could do it again this year.